Calculus Examples

$f (x) \cdot (4 \ln (3 x^{2} - 5 x))$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Move $4$ to the left of $f (x)$ .

$\frac{d}{d x} [4 \cdot f (x) \ln (3 x^{2} - 5 x)]$

Step 1.2

Since $4 f (x)$ is constant with respect to $x$ , the derivative of $4 f (x) \ln (3 x^{2} - 5 x)$ with respect to $x$ is $4 f (x) \frac{d}{d x} [\ln (3 x^{2} - 5 x)]$ .

$4 f (x) \frac{d}{d x} [\ln (3 x^{2} - 5 x)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 3 x^{2} - 5 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x^{2} - 5 x$ .

$4 f (x) (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x^{2} - 5 x])$

Step 2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$4 f (x) (\frac{1}{u} \frac{d}{d x} [3 x^{2} - 5 x])$

Step 2.3

Replace all occurrences of $u$ with $3 x^{2} - 5 x$ .

$4 f (x) (\frac{1}{3 x^{2} - 5 x} \frac{d}{d x} [3 x^{2} - 5 x])$

Step 3

Differentiate.

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Step 3.1

Combine $\frac{1}{3 x^{2} - 5 x}$ and $4$ .

$\frac{4}{3 x^{2} - 5 x} f (x) \frac{d}{d x} [3 x^{2} - 5 x]$

Step 3.2

Combine $\frac{4}{3 x^{2} - 5 x}$ and $f (x)$ .

$\frac{4 f (x)}{3 x^{2} - 5 x} \frac{d}{d x} [3 x^{2} - 5 x]$

Step 3.3

By the Sum Rule, the derivative of $3 x^{2} - 5 x$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 5 x]$ .

$\frac{4 f (x)}{3 x^{2} - 5 x} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 5 x])$

Step 3.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$\frac{4 f (x)}{3 x^{2} - 5 x} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x])$

Step 3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{4 f (x)}{3 x^{2} - 5 x} (3 (2 x) + \frac{d}{d x} [- 5 x])$

Step 3.6

Multiply $2$ by $3$ .

$\frac{4 f (x)}{3 x^{2} - 5 x} (6 x + \frac{d}{d x} [- 5 x])$

Step 3.7

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$\frac{4 f (x)}{3 x^{2} - 5 x} (6 x - 5 \frac{d}{d x} [x])$

Step 3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{4 f (x)}{3 x^{2} - 5 x} (6 x - 5 \cdot 1)$

Step 3.9

Multiply $- 5$ by $1$ .

$\frac{4 f (x)}{3 x^{2} - 5 x} (6 x - 5)$

Step 4

Simplify.

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Step 4.1

Reorder the factors of $\frac{4 f (x)}{3 x^{2} - 5 x} (6 x - 5)$ .

$(6 x - 5) \frac{4 f (x)}{3 x^{2} - 5 x}$

Step 4.2

Factor $x$ out of $3 x^{2} - 5 x$ .

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Step 4.2.1

Factor $x$ out of $3 x^{2}$ .

$(6 x - 5) \frac{4 f (x)}{x (3 x) - 5 x}$

Step 4.2.2

Factor $x$ out of $- 5 x$ .

$(6 x - 5) \frac{4 f (x)}{x (3 x) + x \cdot - 5}$

Step 4.2.3

Factor $x$ out of $x (3 x) + x \cdot - 5$ .

$(6 x - 5) \frac{4 f (x)}{x (3 x - 5)}$

Step 4.3

Multiply $6 x - 5$ by $\frac{4 f (x)}{x (3 x - 5)}$ .

$\frac{(6 x - 5) (4 f (x))}{x (3 x - 5)}$

Step 4.4

Move $4$ to the left of $6 x - 5$ .

$\frac{4 (6 x - 5) f (x)}{x (3 x - 5)}$